Rotating Chemical Waves in the Gray-Scott Model

DigitalCommons@WPI

Mathematical Sciences Faculty Publications

Department of Mathematical Sciences

2-1-1992

Rotating Chemical Waves in the Gray-Scott Model

William W. Farr

Worcester Polytechnic Institute, [email protected]

M. Golubitsky

Follow this and additional works at:

http://digitalcommons.wpi.edu/mathematicalsciences-pubs

Part of the

Mathematics Commons

This Article is brought to you for free and open access by the Department of Mathematical Sciences at DigitalCommons@WPI. It has been accepted for inclusion in Mathematical Sciences Faculty Publications by an authorized administrator of DigitalCommons@WPI.

Suggested Citation

Farr, William W. , Golubitsky, M. (1992). Rotating Chemical Waves in the Gray-Scott Model.SIAM Journal on Applied Mathematics, 52(1), 181-221.

ROTATING CHEMICAL WAVES IN THE GRAY-SCOTf MODEL*

W. W. FARRt AND M. GOLUBITSKYI

Abstract. A set of reaction-diffusion equations is considered, known as the Gray-Scott model, defined on a circle, and the stability of rotating wave solutions formed via Hopf bifurcations that break the circular 0(2) symmetry is investigated. Using a hybrid numerical/analytical technique, center manifold/normal form reductions are performed to analyze symmetry-breaking Hopf bifurcations, degenerate Hopf bifur- cations, and Hopf-Hopf mode interactions. It is found that stable rotating waves exist over broad ranges of parameter values and that the bifurcation behavior of this relatively simple model can be quite complex, e.g., two- and three-frequency motions exist.

Key words. Hopf bifurcation, 0(2) symmetry, reaction-diffusion equations, normal form reduction, rotating (traveling) waves, mode interactions, Gray-Scott model

AMS(MOS) subject classifications. 58F39, 58F36, 58F14, 35K57, 35B32

1. Introduction. In this paper we study numerically degenerate Hopf bifurcations of a reaction-diffusion model posed on a circle,

(1.1 ) ut = _{Dug +f(u, a),}

where u is a 2X-periodic function of ; with values in R', D is a positive-definite diagonal matrix of diffusion coefficients,

f

Our goal is to show that Hopf bifurcations to rotating wave solutions can be expected to occur in any sufficiently realistic model of the B-Z reaction. Of course, these waves need not be stable at bifurcation and might only be observable at finite amplitude after a secondary bifurcation where stability is restored. In this paper we show numerically that both stable and unstable Hopf bifurcations to rotating waves do occur in one of the simplest models for autocatalytic reactions on the annulus-the Gray-Scott model. In addition, we show that in this model the unstable branches of rotating waves can regain stability through secondary bifurcations. It is important to note that the numerical techniques we have used can, in principle, be applied to any

* Received by the editors February 5, 1990; accepted for publication (in revised form) November 8, 1990. The research was supported in part by Texas Advanced Research Program grant ARP-1100, and by National Science Foundation/Defense Advanced Research Projects Agency grant DMS-8700897.

t Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts 01609.

1 Department of Mathematics, University of Houston, Houston, Texas 77204-3476. 181

of the more realistic models. Moreover, as these models all contain at least as many parameters and equations as the Gray-Scott model, it seems likely that they will contain a solution structure at least as complicated as the one we describe for the Gray-Scott model.

We present the results of these numerical calculations to suggest an alternative to the methods of excitable media [Tyson and Keener, 1988], [Maginu, 1985], [Feroe, 1986], [Ellphick, Meron, and Spiegel, 1990] for finding periodic solutions to reaction- diffusion equations. We are interested, in particular, in the structure of solution sets that should be common to all such systems of equations. In particular, the annular geometry forces the existence of 0(2) symmetry; and hence, the expectation of the existence of symmetry-breaking Hopf bifurcations to periodic solutions known as rotating waves. Since we consider the symmetry of the equations to be of paramount importance, we abstract the annulus to a circle. In the context of reaction-diffusion equations on the line with periodic boundary conditions (the circle), it is impossible to obtain rotating wave solutions from stable Hopf bifurcations with fewer than three equations. For this reason we have attempted our numerical calculations on a system with that minimum number of components: the Gray-Scott model.

The Gray-Scott model [Gray and Scott, 1983] is the simplest model consistent with chemical principles that is known to exhibit temporal oscillations in a continuous stirred reactor. The chemical mechanism for this autocatalytic model consists of the two reversible reactions

(1.2) A+2B -3B, B ->C,

which form the core of models like the much-studied Brusselator. Though this model is too simple to hope to describe B-Z chemistry quantitatively, it mimics some of the behavior in a continuous stirred reactor [Scott and Farr, 1988]. Another positive feature of this model is that no species are assumed to be artificially held constant, i.e., the "pool chemical" assumption is not used.

The main mathematical technique in this paper is that of center manifold/normal form reduction. Because we consider degeneracies in the spirit of [Golubitsky and Roberts, 1987] and [Chossat, Golubitsky, and Keyfitz, 1986], we must deal with center manifolds of dimensions as large as six and normal forms computed out to as high as fifth order. The resulting calculations are formidable and beyond the reach of hand calculation, so we have adopted a hybrid approach: the bifurcation formulae are obtained analytically but evaluated numerically. Obtaining the correct formulae is usually a difficult and time-consuming task in itself, but with the aid of the results in [Cushman and Sanders, 1986] and [Elphick et al., 1987] we have devised a systematic procedure for doing so that is easily checked using elementary combinatorics. Details of this procedure, which should extend easily to other types of bifurcations, are found in the Appendix.

We note that a very similar hybrid procedure was used before in [Labouriau, 1985, 1989], [Farr, 1986], and [Farr and Aris, 1987] to analyze degenerate Hopf bifurcations in models described by ordinary differential equations without symmetry. One minor difference is that these earlier investigations used the Lyapunov-Schmidt reduction, as described in [Golubitsky and Langford, 1981].

As an aside, we note that an alternative route to performing center manifold/nor- mal form calculations involves the use of symbolic manipulators such as MACSYMA or Maple. Several monographs are available that detail this approach, for example, [Rand and Armbruster, 1987]. Although the Maple package was used to check some of the calculations in ? 4, we chose not to use it for the reduction calculations for the

following reasons. First, a bifurcation analysis must start with a basic solution whose properties and stability are known. In the case of a reaction-diffusion system on a circle, this basic solution is constant. However, for the (relatively simple) Gray-Scott model, this basic solution cannot be found analytically. Because of this, all of the subsequent calculations must of necessity be done numerically. Given this fact, a completely numerical code is likely to be much faster. Second, we are interested in developing methods that will be useful for analyzing more complicated models and domains. If anything, these extensions will involve considerably more in the way of numerical computation and tip the balance even further away from the type of problem suitable for symbolic manipulation packages. Finally, symmetry-breaking bifurcations, and especially interactions between symmetry-breaking modes, lead to center manifolds of relatively high dimension. This leads to bookkeeping difficulties, especially when higher-order terms are needed. The algorithmic approach detailed in the Appendix describes our attempts to minimize the amount of work that must be done to calculate the normal form up to some given order.

The organization of this paper is as follows. In ? 2 we summarize the experimental results of [Noszticzius et al., 1987] that motivate our work, briefly state our results for the Gray-Scott model, and relate our work to that of earlier investigators. In ? 3 background material on Hopf bifurcation and Hopf-Hopf mode interactions with 0(2) symmetry for systems of reaction-diffusion equations posed on a circle is pre- sented, which covers the main issues and summarizes the needed results. The Gray-Scott model is described in ? 4, where we consider steady-state solutions that are constant on the circle. A linear stability analysis of these solutions allows us to determine which types of bifurcations occur and to locate them precisely. Finally, in ? 5 we present the results of our nonlinear analysis for this model. By calculating coefficients in the normal form corresponding to a bifurcation point and using the material in ? 3, we are able to describe locally the stability of emerging solutions, for example, rotating waves. In particular, we describe regions of parameter space for which stable rotating waves appear via a primary Hopf bifurcation that breaks the 0(2) symmetry. By computing the normal form up to cubic order for a mode interaction between two Hopf bifurcations (one 0(2)-invariant and one that breaks the 0(2) symmetry) as in [Chossat, Golubitsky, and Keyfitz, 1986] we find a second region where the primary bifurcation is to stable 0(2)-invariant periodic solutions, but rotating waves issuing from a second Hopf bifurcation become stable via a secondary bifurcation. Thus we find two periodic solutions that are simultaneously stable. To our knowledge, this is the first time that the six-dimensional 0(2) Hopf-Hopf interaction has been implemented numerically. We also briefly describe additional degeneracies that appear in this model: some have been analyzed (degenerate Hopf bifurcation with 0(2) symmetry [Golubitsky and Roberts, 1987] and Takens-Bogdanov bifurcation with 0(2) symmetry [Dangelmayr and Knobloch, 1987]), while others (0(2) Hopf-invariant limit point, Takens- Bogdanov-0(2) Hopf), have not. The overall picture is considerably complicated when compared to the CSTR dynamics.

2. Motivation and summary. The Belousov-Zhabotinskii (hereafter B-Z) reaction first came to the attention of researchers as an example of a chemical system exhibiting spontaneous temporal oscillations in well-stirred closed (i.e., no mass transport across the boundaries) [Field and Burger, 1985] and continuous flow reactors [Field and Burger, 1985], [Maselko and Swinney, 1986]. The occurrence of spatial patterns and waves in unstirred reactors was also noted earlier [Winfree, 1972], and has been of considerable interest, e.g., [Field and Burger, 1985]. In this paper we are primarily

concerned with spatio-temporal pattern formation in unstirred vessels. The early experiments in this area were almost exclusively carried out in closed vessels, in which a chemical system inexorably relaxes to a spatially uniform, constant equilibrium state, so the patterns observed were really transients. More recent experiments have attempted to overcome this problem. For example, the reactor of [Tam et al., 1988], which allowed sustained spiral waves to be observed, is in the form of a thin disk of gel (to suppress bubble formation and convection), which is fed continuously from beneath by a network of capillary tubes that communicate with a stirred, continuously fed reservoir. The experiments we are most interested in, however, are those of [Noszticzius et al., 1987]. A schematic of their apparatus is shown in Fig. 1. It consisted of a thin annular ring of inert gel sandwiched between impermeable boundaries top and bottom. The inner and outer edges of the annulus are exposed to stirred reservoirs continuously fed with the two components of the B-Z reaction. One advantage of this reactor, over the disk reactor of [Tam et al., 1988] as far as analysis is concerned, is that no reaction occurs in the reservoirs.

FIG. 1. Schematic of annular gel reactor used by [Ncszticzius et al., 1987].

We now describe the experimental results of [Noszticzius et al., 1987]. The outer reservoir contains the sulfuric acid and potassium b romate component, while the malonic acid and the ferroin catalyst are fed to the inner reservoir. When the reactor is started up, a circular front of ferroin advances froni the inside. As it approaches the outer edge, the front becomes more irregular. Eventually, a state develops with several wave sources (pacemakers) distributed randomlly around the outer edge, as illustrated in Fig. 2. The number of wave sources is irregular, as are their frequencies. The source for these pacemakers is not well understood, but they are thought to be related to local inhomogeneities, e.g., foreign objects, and it is thought that they can be removed by refining experimental techniques [Noszticzius et al., 1987]. It must be emphasized that these states are not accessible via the local bifurcation theory we will present in the next section. The experimenters, however, are able to perturb these states systematically and obtain stable rotating wave solutions similar to the one with eight wavefronts shown schematically in Fig. 2. They further state that the rotating waves are stable if the number of wavefronts lies between 6 and 25; below this range pacemakers appear spontaneously, and above it the waves interact destructively. From our point of view, the most striking feature of these states is their regularity and resultant high degree of symmetry. In fact, they possess spatio-temporal symmetry. This concept is best illustrated by example. Consider a state with eight equally spaced

Sink

Pacemaker source

FIG. 2. Top: single source and sink for pacemaker-induced chemical waves. Bottom: stable eight wavefront

rotating wave.

wavefronts. The state is periodic in time, but also in space: rotation of the solution through an angle of 2ir-/8 is the same as letting the solution evolve through one temporal period.

Turning to the concept of symmetry, we consider the group 0(2) generated by the rotations and reflections in the plane. It turns out that, in a sense to be made precise in ? 2, the reaction-diffusion equations are invariant under 0(2). We now describe how the experimental results of [Noszticzius et al., 1987] are consistent with a Hopf bifurcation that breaks that 0(2) symmetry. It is well known [Ruelle, 1973], [Schecter, 1976], [van Gils and Mallet-Paret, 1984] that two families of periodic solutions originate from such a point: standing waves and rotating waves. The first family will not be very important in this work, but the rotating waves are candidates for explaining the experimental results since they have the exact spatio-temporal symmetry described above.

To simplify the presentation and the calculations, we now consider reaction- diffusion equations defined on a circle as an approximation to a thin annulus. Such equations have the following form:

(2.1) ut= Duc+f(u, a),

where u is a 2irr-periodic function of ; and has values in Rn, D is a positive-definite diagonal matrix of diffusion coefficients, and a is a vector of parameters. The action of 0(2) on the circle is generated by

(2.2) Rotation by qi: e+ qi mod 2ir,

and the action of 0(2) on the system of reaction diffusion equations is generated by

(2.3) Rotationbyq1: u(;)->u(;-),

Reflection: u(?) - u(->).

We want to use bifurcation theory to analyze (2.1); a first requirement is a solution from which to bifurcate. Note that we have not required the existence of a trivial solution to (2.1), and the model we consider does not have one. Their place in the theory, however, is taken by solutions that are invariant under the action of 0(2). Such solutions are easily seen to be constants, the solutions of the set of equations f(u, a) = 0. The next step is to linearize (2.1) about one of these invariant solutions and decompose the stability analysis into an infinite sequence of finite-dimensional problems by using Fourier modes. That is, we consider linear stability of the invariant solution to disturbances of the form v ei"' eAt, where v E Rn and m = 0, ?E1, +2,*

The resulting finite-dimensional eigenvalue problem is

(2.4) Av = (A - _m2D)v,

where A is the Jacobian of f(u, a) at the invariant state in question. It is clear from (2.4) that the eigenvalues for m are duplicated for -m. This is a direct result of the 0(2) symmetry. We call an invariant steady-state of (2.1) a bifurcation point if it has an eigenvalue with zero real part for at least one value of m. If m $ 0 for one of the critical eigenvalues, we call the bifurcation point symmetry-breaking. We note that a decomposition into a direct sum of finite-dimensional subspaces is guaranteed by the theory of representations [Kirillov, 1976], but this decomposition is particularly con- venient for calculations, as we will see below. We also note that since D is positive- definite the eigenvalues eventually have negative real parts for all sufficiently large m.

In most chemical systems of interest, the diffusion coefficients are nearly equal so the matrix D is close to a scalar multiple of the identity. If this is exactly true, then the eigenvalues of (2.4) for arbitrary m are related to those for m = 0 by translation, which can be seen easily by substituting D = dI into (2.4). An important observation for our work on Hopf bifurcation is that, for equal diffusion coefficients, a symmetry- breaking Hopf bifurcation can never be the primary bifurcation point. This follows from (2.4) since, if there is a pure imaginary pair of eigenvalues for some m > 0, then there are complex conjugate pairs of eigenvalues with positive real part (= n2d) for all n, 0- n - m. Thus to find stable rotating waves we must either have unequal diffusion or some mechanism by which the rotating waves can become stable when they are not the primary bifurcation. In our work with the Gray-Scott model, we encounter both of these cases.

We now briefly describe the Gray-Scott model and the results we have obtained. Technical details are given in ?? 3-5 and the Appendix. The model equations are

x, = DIx,-xy2 +aly3+A(1-x),

(2.5) yt = D2yC + Pxy2 -a1Y3-y Y+a3Z+A(y2-y), z, = D3zC+y-a3z+A(y3-z)

and are essentially those of [Balakotaiah, 1987] with diffusion terms added. The parameters appearing in (2.5) have physical meaning as follows: ,3 represents a forward reaction rate coefficient, a1 and a3 are reverse reaction rate coefficients, A represents a mass-transfer coefficient, and Y2 and y3 are ratios of the feed concentrations of

invariant solutions can be reduced to solving a single nonlinear equation that is cubic in y. Analytic solution is not possible, but the methods of [Golubitsky and Schaeffer, 1985] are used in [Balakotaiah, 1987] to classify all of the possible bifurcation diagrams for the invariant steady-states. For any particular parameter values the bifurcation diagram can be generated numerically using a standard path-following technique [Doedel, Jepson, and Keller, 1984].

Linear stability calculations must also be done numerically, and standard path- following techniques exist [Spence and Jepson, 1984]. Because the model is only three-dimensional, however, it is simpler to use the characteristic equation. Conditions for eigenvalues with zero real part can be formulated simply in terms of the characteristic equation coefficients and solving for particular configurations, for example, finding Hopf bifurcation points for a particular value of m, is relatively easy. However, to obtain local stability results from a center manifold/normal form reduction we really need information on all of the eigenvalues. This can be obtained as follows. We will show below that if A is small enough, all of the eigenvalues have negative real parts. Hence, we can determine changes in stability simply by monitoring the signs of the conditions for eigenvalues with zero real part. In principle, we would have to do this for all values of m, but in practice we only have to monitor a finite number of modes because the eigenvalues eventually have negative real parts for large m.

Some typical results from the analysis are shown in Fig. 3. The parameter values were chosen so that there is a unique invariant steady-state and the only bifurcations occurring are Hopf bifurcations for m =0 and m = 1. The invariant steady-state is initially stable. First, a pair of m = 0 Hopf points appear and separate, then a pair of m = 1 Hopf points appears between the m = 0 Hopf points and, finally, the m = 1 Hopf point at the smaller value of A passes through the m = 0 Hopf point and becomes the primary bifurcation point. The places where a pair of Hopf points appear are our first examples of degenerate Hopf bifurcation [Golubitsky and Langford, 1981], [Golubitsky

1 2

y y

* m-O Hopf point o m-1 Hopf point

3 ₄

Y y

FIG. 3. Typical linear stability results for Gray-Scott model. (1) Stable invariant steady-state. (2) Two

m = 0 Hopf points appear. (3) A pair of m = 1 Hopf points appears. (4) The leftmost m = 1 Hopf point becomes a primary bifurcation point by passing through the m = 0 Hopf point. Main branch is steady-state; bifurcating branches correspond to periodic solutions.

and Roberts, 1987], and the crossing of the m = 0 and m = 1 is a mode interaction [Chossat, Golubitsky, and Keyfitz, 1986]. By computing the normal form vector field for each Hopf point we can determine the local stability of the periodic branches, and this is also shown in Fig. 3.

By computing the six-dimensional normal form for the interaction of the m = 1 and m = 0 Hopf points up to cubic order and using the results in [Chossat, Golubitsky, and Keyfitz, 1986] we can obtain more information on how the stabilities change. Details are given below, but the local diagrams are as shown in Fig. 4. There we see that secondary bifurcations to quasi-periodic solutions (which are not stable) occur such that the rotating waves and the invariant periodic branch each become stable when they are not the primary bifurcation. One interesting result is that the rotating waves and the invariant periodic solutions are simultaneously stable.

-~~~~

1 2

2

FIG. 4. Sample local bifurcation diagrams for the six-dimensional m = 0, m = 1 Hopf-Hopf mode interac-

tion. The branch of rotating waves is labeled 2.

In ? 5 we present more results by varying an additional parameter as well. It turns out that the sequence we have presented above holds in a fairly large region of parameter space, but we also encounter additional complications. For example, limit points can appear on the invariant steady-state branch via a hysteresis bifurcation. This leads to further complications in the form of Takens-Bogdanov singularities involving the invariant limit points and invariant Hopf points as well as interactions of the m = 1 Hopf points with the invariant limit points. A codimension three singularity occurs when we have an invariant Takens-Bogdanov simultaneously with an m = 1 Hopf point. By changing the values of the diffusion coefficients, Hopf modes for any m can appear, although only the m = 0, 1, or 2 modes are primary bifurcation points in our investigations. Since the diffusion coefficients in the Gray-Scott model cannot be related to the experiments, we cannot predict which m will actually appear in the experiments. Earlier investigators also interested in rotating waves in circularly symmetric geometries include [Auchmuty, 1979, 1984] and [Erneux and Herschkowitz-Kaufman, 1977, 1979a, 1979b], who both considered the Brusselator model on a two-dimensional disk. This model is nice to work with because it has a trivial solution and only two

chemical species, so analytical calculations are feasible. Unfortunately, as we will see below, for Hopf bifurcation in a model with two species the primary bifurcation is always to invariant periodic solutions, and bringing together Hopf points corresponding to different modes is impossible for nonzero diffusion coefficients. Thus local analysis cannot be used to find stable periodic solutions that have broken the 0(2) symmetry. [Erneaux and Herschkowitz-Kaufman, 1979a, 1979b] use numerical simulation in addition to local bifurcation theory and find that both standing and rotating waves can be stable. In our analysis these states can be found by secondary bifurcation.

3. Hopf bifurcation with 0(2) symmetry and mode interactions. In this section we briefly summarize concepts concerning Hopf bifurcation with 0(2) symmetry and the 0(2) Hopf-Hopf mode interactions. We give normal forms and describe the solutions that appear for each bifurcation, but we do not present details. This material is taken, for the most part, from [Golubitsky, Stewart, and Schaeffer, 1988] but we refer readers also to [Golubitsky and Roberts, 1987] and [Chossat, Golubitsky, and Keyfitz, 1986] for further details.

We first consider symmetry-breaking Hopf bifurcation with 0(2) symmetry. In our application this occurs when there are (nonzero) purely imaginary eigenvalues for some fixed positive value of m. Because the eigenvalues are reproduced for - m, the critical or center subspace is generically four-dimensional. As for Hopf bifurcation without symmetry, it is convenient to use complex coordinates, so we identify the four-dimensional space with (z1, Z2) E C2. We will show in the Appendix how these coordinates are chosen so that they inherit the 0(2) action given by

(3.1) 1(z,, Z2)= (e'qzl_ e- iZ2), K(Z1, Z2) = (Z2, Z1),

where qf is the rotation and K is the reflection. These coordinates were used by [van Gils and Mallet-Paret, 1984]. Because a Hopf bifurcation produces solutions periodic in time, there is an additional phase-shift symmetry [Golubitsky and Stewart, 1985], [Golubitsky, Stewart, and Schaeffer, 1988], that of the circle S', which acts by

(3.2) O(Z1, Z2)= (eiOzl, eioz2).

Details from the group theoretical standpoint of how this phase-shift symmetry arises are in the two previous references, but their result is that the normal form can be chosen to commute with the group 0(2) x S1. This same result appears in the formal characterizations of normal forms of [Cushman and Sanders, 1986] and [Elphick et al., 1987]. Given the actions of 0(2) and S above, it can be shown that a normal form for Hopf bifurcation with 0(2) symmetry can be represented up to any finite order by

(3.3)

(z)

(

where 8 = z212- z 12 and the quantities Pi, q1, r1, and s, are real functions of N=

|z 12+1Z212, A = 62, the bifurcation parameter A, and (in the case of degeneracies) other

real parameters. Generically, there are two types of solutions to (3.3) near the origin: standing waves and rotating waves. In applications, it is important to note that the rotating waves appear in pairs that travel in opposite directions, but according to the theory they are identified as being essentially indistinguishable, since one is related to the other by the reflection K, and thus behave identically. An important feature is that (3.3) can be separated into phase and amplitude equations and the resulting amplitude equations can be analyzed using singularity theory to obtain relatively complete

information on degenerate Hopf bifurcation up to codimension two [Golubitsky and Roberts, 1987].

As we said above, symmetry-breaking Hopf bifurcation occurs in our application when there are purely imaginary eigenvalues for some positive value of m. In the normal form (3.3) no trace of m remains and this deserves some explanation. Details are in [Golubitsky, Stewart, and Schaeffer, 1988] but the idea is that we factor out a cyclic kernel to obtain the standard action. This factoring out adds to the elegance of the theory but conceals the fact that the value of m is crucial to visualizing the appearance of the bifurcating solutions in the application. In the case of rotating waves on a circle, the value of m determines the number of peaks or wavefronts that appear. The linear degeneracies that are of the most interest in this study are what we term Hopf-Hopf mode interactions, that is, bifurcation points where there are purely imaginary eigenvalues for more than one value of m simultaneously. We saw above that for equal diffusion coefficients this cannot happen and, furthermore, that the various modes (if they appear at all) appear in a very ordered fashion. Thus when we vary the diffusion coefficients away from equality, we would expect to see interactions between neighboring modes first. This turns out to be what we find for this model. Since the m = 0 mode is the primary bifurcation for equal diffusion coefficients, we will focus on the m = 0, m = 1 Hopf-Hopf mode interaction, but we will also encounter

the m = 0, m = 2 and m = 1, m = 2 interactions.

A general principle of local bifurcation theory is that it can be used to find stable bifurcating solutions only from the primary bifurcation. A step away from this limitation is the study of mode interactions, that is, situations where by varying one additional parameter we can get the bifurcation points for distinct modes to pass through each other. In such a situation each mode is the primary bifurcation for some parameter values and so can be expected to produce stable branches. A well-known result is that mode interactions can produce secondary or even tertiary bifurcations, which compli- cate the dynamics substantially. Studying these interactions via normal forms permits us to predict considerably more about the behavior of the system than can be determined from studying either mode alone. In our application, we are primarily interested in two types of Hopf-Hopf mode interactions.

The first type is the interaction of an invariant Hopf (m =0) and a symmetry- breaking Hopf (m # 0). The center manifold for this singularity is six-dimensional, and again complex coordinates (zO, zl, z2) are the most convenient. We refer the reader to [Chossat, Golubitsky, and Keyfitz, 1986] for details. They show that if the two frequencies are incommensurate, the normal form commutes with the action of 0(2) x T2 generated by

(zo, zl, z2) = _(zo, ei'zl , e-iz2) 9 fE SO(2) ' 0(2),

(3.4) K(Zo Zl, z2) = (ZO, Z2, Z1),

(09 P) (ZO , Zl , Z2) = ( e "zo e "'zl

.

.

Note that again it is the standard action of 0(2) that appears in (3.4), but the same warning for interpreting the results applies. The normal form for this singularity can be written as

(io\

(3.5) {i,) =(po+iq0) 0) +(p1+iql) z) +(P2+iq2)j6 8 Zl

6 = jz212-

I

TABLE 3.1

Identification of branches for six- dimensional Hopf- Hopf interaction.

Branch Identification 1 invariant periodic 2 rotating wave 3 standing wave 4 two frequency 6 two frequency

For our purposes, we need only consider the cubic order truncation of the amplitude equations

io= ro(po,Ak +po,p3 +poor2+pol(r2+ r2)),

(3.6) ii = r1(pl,,kA +pl,p3, +p1or2 +pjjr2+p12r2), i2= r2(pl,AA +Pl,p13 +ploro+Pl2rl +pllr2)

which is obtained by truncating (3.5) at cubic order, writing zj = rj e 'J, and separating real and imaginary parts. Note that symmetry requires certain terms to equal; this is reflected in (3.6). This is not quite the same as the equations that are used in [Chossat, Golubitsky, and Keyfitz, 1986], but if pos $ 0, Pl,, $ 0, and Po,APi, -Pl,APo,p #0 , (3.6) can be reduced to the case they consider by defining new bifurcation and unfolding parameters that are appropriate linear scalings and combinations of A and ,8. If this can be done and ten additional nondegeneracy conditions involving the coefficients in (3.6) are satisfied, then the results of [Chossat, Golubitsky, and Keyfitz, 1986] allow us to determine the stabilities of the primary branches from each of the Hopf points and also locate and determine stability of branches originating from the secondary bifurcations. For the present study, it was found convenient to repeat the stability calculations without making a preliminary change of A, , coordinates. The results are

.,oB U>-o < _ =F*_ - ta W- \O (W-% \O X

v~ 4)eo

~

~

I? I I14-1 P14 I--, S V Ce e X I t _ C n 5 C 0 *$ . cd~~~~~~~~~~~~~~~~~~~~~~~~~~~~4 N N. 0 21 2 0 40 0 Q -~4A - Q N N

F Pt} j tR4'a :::u e; I C W < W tt - Nto C _ t< _W N- 0 <

0 4) CZa Ca.< t- < no Z + s o= I r 4-1 0

~~~~~~~~~~~~~~~~~~~~~~~~~

~~~~~

summarized in Table 3.2 (based on Table 4.5 in [Chossat, Golubitsky, and Keyfitz, 1986]), with the definitions needed to evaluate the quantities appearing in Table 3.2 given in Table 3.3. The reader should note that Table 3.2 is considerably more useful for applications. The Eij matrix appearing in the tables is similar to one defined in [Melbourne, Chossat, and Golubitsky, 1989], which turns out to be very useful in determining the stabilities of the three primary branches. For example, if the entries in the jth column are all negative, then branch j will be stable when it originates from the primary bifurcation point and will become stable via secondary bifurcations when the other mode bifurcates first.

TABLE 3.3

Definitions of quantities in Table 3.2.

I I Poo 21= (PO,API1OPI,APOO)/PO,A

?12=PII -22 =PI2-PI1

B13 =PII +PI2 E23 = PI I-P12

?31 (PO,APIlOPI,APOO)/PO,A

?32 = _{(Pl,APOI PO,APll)/Pl,A}

?33 [ [2pI,AkpOI PO,A((PII +PI2)]/PlI1A

PI,A PO,AP 1,G PI,APO,,G K = Ek ,=

PO,,A PO,,A

PIoPo,13 PooPi, "G PoiPI,1 PiiPo,13 2poIpI,j6 (Pll +PI2)Po,p

PO,A PI,A PI,A

61 = -12-21 + KE, I E32 e2 = '13 -31 + K-1 I 1 33 A4 = -(?1271 + KEII7?2)/6i A6 = -(?1371 + K1ii7?3)/e2

It can also be shown that seven of the ten nondegeneracy conditions of [Chossat, Golubitsky, and Keyfitz, 1986] are equivalent to requiring no nonzero entries in the Eij matrix (the careful reader will have already noted that E21 = ?31 and ?22 = -E23, So

there are only seven independent entries). Their remaining three conditions are provided by requiring nonzero values for P12, det 4, and det 6. Note that since (3.6) is more

general than the normal form considered in [Chossat, Golubitsky, and Keyfitz, 1986] we additionally require nonzero values for Po,A, Pi,A and Po,AP1,p, -Pj,xPko,P (=Po,A8AxP),

bringing the total number of nondegeneracy conditions to 13. If these conditions are satisfied, then the direction of branching and stability of each of the five primary and secondary branches is determined by the cubic order truncated normal form (3.6). In addition, any secondary bifurcation points that appear for a fixed value of X3 #0 will do so at distinct values of A. If any one of them is not satisfied, then further analysis must be done.

The final singularity of interest is the interaction of two 0(2) symmetry-breaking Hopf bifurcations, which has an eight-dimensional center manifold. Complex coordi- nates (zI, z2, z3, z4) e C2G C2 are again the most convenient. When the frequencies are nonresonant, the relevant symmetry group is again 0(2) x T2 as it was for the six-dimensional interaction, but the action is more complicated since 0(2) does not act trivially on any of the coordinates, as it did on zo in the previous case. Using the same notation as above, the relevant action is generated by

fr(z, Z2, Z3, Z4)= (eik zi, e-ik z2, ein'z3, e-inqI4), Z f E SO(2) c 0(2),

(3.7) K(Z1, Z2, Z3, Z4) = (Z2, Z1, Z4, Z3),

The integers k and n, k < n, that appear in (3.7) are the values of m to which the two Hopf points correspond. In our application we will be interested in the case where k = 1 and n =2. More generally, the values of k and n are obtained by removing common factors between the two values of m and factoring out by a cyclic kernel, but we do not need this generality. We do note, however, that in the general case it is not possible to assume that 0(2) acts via its standard representation on (z,, z2), even though this turns out to be possible for the case we consider here.

The normal form for the eight-dimensional Hopf-Hopf interaction is developed in detail in [Chossat, Golubitsky, and Keyfitz, 1986]. We merely reproduce the final result:

il = (P1 + iq1)z, + (r, + iS)n1Zn (Z3Z4) k,

(38= (P2+ iq2)Z2+ (r2+ iS2)zl( 2 3z4),

(3.8) Z3-= p3 + iq3)z3 + (r3 + iS3)(Zl 32) Z3 4 9

Z = (Pp4+ iq4)z4+ (r4+ _iS4)(51Z2) Z3 Z4 ,

where the real functions pj, q;, rj, sj, j = 1, 4 depend on pi = _IZ2, i = 1, 4 and the real and imaginary parts of a = _{(z1I2)n(53z4)k.} In addition, the reflection K imposes relations between the first and second and third and fourth components of the vector field, which are easiest to describe as follows. Suppose that we represent (3.8) by ij =

Fj(zl, z2, Z3, Z4), j = 1, 4. Then the restrictions that are placed on the vector field by K

are

F2(zl, Z2, Z3, Z4) =F1(K(Z1, Z2, Z3, Z4)) =F1(Z2, z1, Z4, Z3),

(3.9)

F4(Zl , Z2,9 Z3,9 Z4) = F3(K (Zl1, Z2!, Z3,, Z4)) = F3(Z2. Zl,, Z4, Z3)-

A major difference from the six-dimensional normal form is that the eight-dimensional normal form cannot be separated into phase and amplitude equations, so the analysis is correspondingly more difficult. Also, adding to the complication in the eight- dimensional case are the four primary branches and the eight types of two- and three-frequency secondary branches that appear in this interaction. In [Chossat, Golubitsky, and Keyfitz, 1986] the branches that can appear in this interaction are classified, and conditions that allow the direction of bifurcation and stability of each branch to be determined are derived. Fortunately, in this study we are concentrating on the rotating waves, and it turns out that their branching and stability are determined at third order, including the behavior of the secondary branches that generically appear on the two branches of rotating waves. Furthermore, the normal form truncated at cubic order does separate into phase and amplitude equations so we can proceed as for the six-dimensional interaction and write our truncated normal form amplitude equations as il _{= rl(plkAk} + pi,pG3 _{+p, 4r2} +p12r2 +p13r2 +p14r2), (3.10) _r2 = _{r2(pl AA} +Pl,03,8 +p12r2 +pllr2 +p14r 2 +p13r2), i3 = r3(p3,Ak +p3,f3P +P31ri +p32r2+p33r3+p34r4), r4= r4(p3,kA +P3,,83/ +P32rl+P31r2+p34r3+p33r4).

Again, note the terms forced to be equal by symmetry. Following the classification of solutions in [Chossat, Golubitsky, and Keyfitz, 1986], we can construct Table 3.4, which provides identifying numbers for the four primary branches and the two secon-

TABLE 3.4

Branch identification for eight- dimensional Hopf-Hopf interaction.

Branch Identification 1 k-rotating wave 2 k-standing wave 3 n-standing wave 4 n-rotating wave 7 two frequency 8 two frequency

dary branches, which generically bifurcate from the rotating waves. We also construct Table 3.5, which gives the branching equations and eigenvalues for these six branches. Table 3.6 provides the definitions of the quantities in Table 3.5. Using the results in the three tables, in ? 5 we will be able to determine local bifurcation diagrams for the eight-dimensional Hopf-Hopf interaction, but only in a limited sense since we will ignore' the branches of standing waves and their secondary bifurcations. It will turn out, however, that for the parameter values we have chosen, the rotating waves are the only stable solutions and the deficiency in our analysis will not be important.

It turns out that even for our limited analysis of the truncated normal form (3.10) there are 23 nondegeneracy conditions, i.e., these conditions must be satisfied for the direction of branching and stability of the four primary branches and two secondary branches we are considering to be determined at third order. Twelve of these conditions come from requiring that all of the rij matrix entries be nonzero, Pl,A $ 0 and P3,A $ 0 follow from requiring strict eigenvalue crossing, and the remaining nine conditions are derived by requiring distinct values of A (for 3 $ 0) for all of the primary and

secondary bifurcation points. These conditions are summarized in Table 3.7.

4. Invariant steady-states and linear stability. In this section we want to establish in detail the structure and linear stability of the 0(2)-invariant solutions. Our main interest continues to be symmetry-breaking Hopf bifurcations, but the preliminary material in this section lays the foundation for our later results. The techniques are singularity theory [Golubitsky and Schaeffer, 1985] and the global extensions due to [Balakotaiah and Luss, 1984] to determine the local structure of the invariant solutions and straightforward manipulation of the characteristic equation to determine stability. The invariant steady-states are obtained by removing the diffusion terms from (2.5) and studying the steady-states of the resulting set of ordinary differential equations

x= =-2xy 2+ _{tly 3+A(1 -x),}

(4.1) j = 3xy2 _al y3-y+a3z+ A(y2-Y),

= y-a3z + _A(Y3-z).

These equations will be referred to below as the CSTR equations, since they describe the behavior of the Gray-Scott model in a continuous stirred tank reactor. As described in [Balakotaiah, 1987] the steady-states of (4.1) can be found from a single equation, cubic in y, that is obtained by setting the right-hand side of (4.1) to zero and eliminating the variables x and z. Furthermore, as shown in [Balakotaiah, 1987], the structure of the solutions of this cubic equation can be completely characterized via the singularity theory methods of [Golubitsky and Schaeffer, 1985] and the extensions in [Balakotaiah

cl~~~~~~~~~~~~~~~~~~~~o o 000 (0~~~~~~~0 (0 00 0 coo~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 0 o~~~~~~~~~~~~~~~~~~o C;- .0 ~~~~COO (Co. N N ~~~~~~~~~~~~~~c c q q e r r N Nc 0 (0 N (0 N *~~~~~~~~~~~~~~~~~~~~~~~~~~~ N-

~

~~~~

~~~~~~~

~

~ ~ ~ ~ ~ ~

~ ~ ~~ ~~

TABLE 3.6

Definitions of the quantities in Table 3.5.

?,l1 = 2p, I F?21 P12 PII

?12 = 2(P 1I+P12) ?22 2(PII IPI2)

?13 = 2(P33 +P34) ?23 = 2(P33 P34)

?14 = 2P33 ?24 P34 P33

831 = (Pl,,A P31 P3,A PI )/Pl,,A

?32 = [PI,A (P31 +P32) P3,A (Pl I +PI2)]/PI ,A ?33 = [P3,A (PI3 +PI4) -Pl,A (P33 +P34)]/P3,A ?34 = (P3,A P14 Pl,AP33)/P3,A

?41 = _{(PI,A P32 P3,A PI )/Pl,,A}

?42 = [Pl,A(P31 +P32) -P3,A(Pl 1 +PI2)]/PI,A ?43 = [P3,Ak(P13 +P14) _PI,A(P33 +P34)]/P3,A ?44 = (P3,AP13 Pl,AkP33)/P3,A

PI,AP3,13 PI,AP3,6 _K P3,A

PI,A P1,A

PI IP3,,1-P31PI,, _k8 P32PI, -PI IP3,13 Pl IP3,A P31PI,A P32PI,A PI IP3,A

A _ P13P3,1 _P33P,J3 _{A =P14P3,-P33P1,,3}

47 -48

PI3P3,A P33PI,A P14P3,A P33PI,A

TABLE 3.7

Nondegeneracy conditions for the eight- dimensional Hopf-Hopf interaction.

1-12 ?ijA 0 i j = 1, 4 13 P1,A : 0 14 P3,A : ? 15 EAP : ? 16 PlIP33-P31PI3 0 ? 17 PI IP33-P32P14A +0 1 8 P 1P33-P3 IP14 0 0 19 _{PI IP33 -P32P13} 0 ? 20 P13:0 21 P14A +0 22 P31:0 23 P32A0

and Luss, 1984]. For the present study we take A as our distinguished bifurcation parameter, 8 and Y2 as free parameters, and fix the values a1 = 0.1, a3= 0.002, and

y3= 0.4. Giyen these values, the nonpersistence varieties and the corresponding steady-

state bifurcation diagrams in the parameter regions of interest are shown in Fig. 5. Actually, the two steady-state limit points that occur in region 4 only enter tangentially into our analysis and can be ignored most of the time, so we can think of (4.1) as possessing a unique invariant steady-state. The numerical values on the axes correspond to sets B and C; an analogous plot for set A is qualitatively similar.

The next step is to study the linear stability of the invariant steady-states. The Gray-Scott model is simple enough so that this can be done using the characteristic

y y 0.14 1 H 0.12 720.1 4v 0.04 0.02 0 0 0.02 0.04 0.06 0.08 0.1 0.12 1/fl

FIG 5. Top: bifurcation diagramsfor invariant steady-states. Numbers refer to open regions below. Bottom: nonpersistence variety H for invariant steady-states. Bifurcation variety B is the boundary of the shaded region.

equation of the linearization of (2.5) about an invariant steady-state. At any invariant steady-state, the characteristic equation can be solved for the eigenvalues for each value of m. Clearly, we need only consider nonnegative values of m when determining linear stability, and, in fact, we will only need to consider a finite range for the values of m. However, it is not necessary to solve for all the eigenvalues if we are only interested in changes in stability. The idea is that we start our stability determination at some point where we know the eigenvalue configuration and then look for changes in stability as we move continuously along the invariant steady-state bifurcation diagram. Generically changes in stability occur via Hopf bifurcation or by a single real eigenvalue crossing through zero, and it is a simple matter to derive necessary conditions from the characteristic equation for each type of bifurcation point. These conditions are actually derived from the Routh-Hurwitz criterion [Levinson and Redheffer, 1970], which gives necessary and sufficient conditions for all of the eigen- values to have negative real parts. This same criterion can be used to show that all eigenvalues have negative real part if A is small but nonzero, giving us a starting point for determining linear stability. From the physical standpoint, having A = 0 means that the system is closed and a continuum of uniform equilibrium states exists. Thus it comes as no surprise that close to equilibrium (i.e., small A) the invariant steady-state is asymptotically stable.

To perform the linear stability analysis we must first choose values for the three diffusion coefficients. Since the Gray-Scott model is not intended to describe any particular physical system, there are no experimentally measured quantities to use as guides as there are for models more closely related to the B-Z system. We are guided, however, by our desire to see if the Gray-Scott model can exhibit behavior consistent with our interpretation of the experiments of [Noszticzius et al., 1987], that is, to choose the parameters so that the symmetry-breaking bifurcations are Hopf bifurcations that lead to stable rotating wave solutions. We would also like to see evidence that

different rotating modes can be simultaneously stable. With these goals in mind, we have chosen the three different sets of diffusion coefficient values shown in Table 4.1 below. For the sake of convenience we also include the values of the other fixed parameters for each set. Note that in set A the value of a3 iS changed to 0.01.

In general terms, we have found that if D -c D2< D3 then Hopf-Hopf mode interactions do occur and symmetry-breaking Hopf points can be primary bifurcation points. In addition, for this ordering of the diffusion coefficients, symmetry-breaking steady-state bifurcations are suppressed and do not interfere with our results. To be more precise, when there is a unique branch of invariant solutions (i.e., Fig. 5(a)), symmetry-breaking steady-state bifurcations do not occur. In parameter regions where there are multiple invariant steady-states, for any fixed value of A there are either one or three such solutions. When there are three solutions we can speak of the middle branch and it is easy to see that this branch is unstable to perturbations with wave number m = 0. Symmetry-breaking steady-state bifurcation points do appear on these middle branches, but if the diffusion coefficients are ordered as above they stay on the middle branches and do not interact with our results. Furthermore, since the middle branch is unstable to perturbations with m = 0, any branch emerging from a symmetry- breaking steady-state bifurcation point on the middle branch of invariant solutions will itself be unstable near the bifurcation point.

The linear analysis for the three sets of parameters shown in Table 4.1 can be conveniently described by grouping sets B and C together, as the results are similar for this pair. Exactly what we mean by similar will be made clear below, but we note that we do not mean identical. Moreover, as we see in ? 5, this similarity does not hold for the nonlinear analysis, and important differences between each of the three sets will emerge. In set A only the m =0 and m = 1 Hopf points and their interaction are important. In sets B and C the m =2 Hopf mode can also be a primary bifurcation point, and six-dimensional interactions between modes m = 0 and each of m = 1 and m = 2 are important, as well as the eight-dimensional m = 1, m = 2 interaction.

The linear analysis for set A is the simplest of the three sets and is a less complicated introduction to how we have chosen to present these results. Essentially, we will build on the results presented in Fig. 5 by adding curves to the parameter space plot, which represent important changes in the linear stability analysis. By important changes we generally mean changes in the number or relative positions of bifurcation points, e.g., degenerate Hopf bifurcations and mode interactions, when these changes produce a change in the primary bifurcation point. These additional curves, together with the steady-state nonpersistence curves we have already presented, divide the parameter plane into open regions. In each of these regions no important qualitative changes in the linear stability analysis occur, so we can represent each region with a single bifurcation diagram showing the relative locations of the Hopf bifurcation points. We will only give explicit results for the modes that actually become primary bifurcation points although we will usually give the largest value of m for which Hopf modes appear.

TABLE 4.1 Three parameter sets.

Set DI D2 D3 a1 a3 Y3

A 0.001 0.001 0.0013 0.1 0.01 0.4 B 0.001 0.0015 0.002 0.1 0.02 0.4 C 0.001 0.001 0.002 0.1 0.02 0.4

In Fig. 6 we have added two curves to those shown in Fig. 5 to describe the important features of the linear stability analysis for set A. These additional curves subdivide regions 1 and 4 into regions la-lc and 4a-4d. The new curves appearing in Fig. 6 can be divided into two classes. The first class describes stability changes for the CSTR version of the equations, that is, we restrict the analysis to the m = 0 subspace. These are shown in Fig. 6, along with the steady-state nonpersistence curves, as the thinner lines. These curves do not depend on the values of the diffusion coefficients and so are exactly the same for sets B and C of diffusion coefficients. We note that the CSTR linear stability results for set A are qualitatively similar to those of the other two sets. The important curves in this first class are labeled DHO, representing a degenerate Hopf bifurcation in which a pair of m = 0 Hopf points appears and T-B represents the Takens-Bogdanov nilpotent double zero eigenvalue singularities. We make no serious attempt to analyze these latter singularities since we consider only Hopf bifurcations in this study, but we do note that they appear and that the DHo and six-dimensional Hopf-Hopf mode interaction curves terminate on the T-B curve. We note that the linear analysis follows the same pattern as shown in Fig. 3, with the invariant steady-state being stable (Fig. 3(a)) in regions la and 4a.

The second class of curves, shown as heavier lines in Fig. 6, represents changes in the linear stability analysis involving symmetry-breaking modes. Only two curves from this class appear in Fig. 6. The curve DH1 represents a-degenerate Hopf bifurcation in which a pair of m = 1 Hopf points appear between the two m =0 Hopf points already present. On the curve labeled H-H a six-dimensional m = 0, m = 1 Hopf-Hopf mode interaction occurs and in the region labeled 4d in Fig. 6, the leftmost m = 1 Hopf point becomes a primary bifurcation point. For our analysis, this is the most important curve in the figure.

Because the parameter space plots (e.g., Fig. 6) are actually projections, curves may appear to cross when, in fact, they do not intersect. To avoid confusion, points of actual intersection are marked with dots in the parameter space plots. If two curves appear to cross but their intersection is not marked in this way, then the two degeneracies represented by the curves do not interact locally. For example, the curve DHo crosses the H curve twice in Fig. 6 but the degenerate Hopf bifurcation and the hysteresis

la DHO

H X IC

4 d 4

r2 _4c

1/J

FIG. 6. Additional curves DHO (m = 0 degenerate Hopf), DH1 (m = 1 degenerate 0(2) Hopf), and H-H (six-dimensional m = 0, m = 1 Hopf-Hopf interaction) in parameter space, arisingfrom linear stability analysis of set A.

singularity occur at different values of A. On the other hand, the H curve and the T-B curve have the two points of intersection marked in the figure. These are degenerate Takens-Bogdanov singularities for which one of the quadratic terms in the normal form vanishes. It is known that the dynamics near such a degeneracy can be quite complicated, but we will not address that here.

Figure 6 summarizes the most important aspects of the linear analysis for set A. Other Hopf modes (at least m = 0 to m = 12) appear and the size of region 4d is actually quite small. However only Hopf modes m = 0 and m = 1 can be primary bifurcation points for set A, so we neglect the other modes since our methods could not be used to obtain stable solutions involving them.

In the shaded region in Fig. 6, additional complications beyond the scope of this paper occur. These complications are caused by multiple invariant steady-states inter- acting with the Hopf bifurcations, producing the Takens-Bogdanov singularities men- tioned above from the m = 0 Hopf points and invariant limit point, symmetry-breaking Hopf point interactions from the other Hopf modes. Further into this region, after the symmetry-breaking Hopf points pass through the invariant limit points, Takens- Bogdanov singularities with 0(2) symmetry [Dangelmayr and Knobloch, 1987] appear. We mention these singularities only in passing to illustrate the complications to be expected even in this rather simple model. In fact, the termination of the H-H curve on the T-B curve in Fig. 6 produces a singularity that has not yet been analyzed. Since it would contain both homoclinic behavior from the m = 0 Takens-Bogdanov singularity and periodic behavior from the symmetry-breaking Hopf bifurcation, homoclinic tangles and weakly chaotic behavior would almost surely be present.

In Fig. 7 we summarize the linear stability analysis for sets B and C of parameters. Two new curves appear that did not appear in Fig. 6. Along these curves, labeled H-H 0-2 and H-H 1-2, Hopf-Hopf interactions occur between the leftmost m = 0 and m = 2 Hopf points (H-H 0-2) and the leftmost m = 1 and m = 2 Hopf points (H-H 1-2). We call the reader's attention to the facts that the leftmost m = 1 Hopf point is still the primary bifurcation point on the H-H 0-2 curve and that the leftmost m = 2 Hopf point is a primary bifurcation point only in the small region labeled 4h. The three Hopf-Hopf interaction curves in Fig. 7 are drawn schematically. In fact, these curves lie very close to the invariant steady-state hysteresis curve H and could not be distinguished from it if the drawing was done to scale. Exact parameter values for some points on the Hopf-Hopf interaction curves will be presented in the next section. Tabulated results for all of the Hopf-Hopf interaction curves are available on request from the authors.

5. Nonlinear analysis and stable rotating waves. In this section we describe the results of our nonlinear analysis of invariant and symmetry-breaking Hopf bifurcations and Hopf-Hopf mode interactions. As the title of this section suggests, our focus is on using the local analysis to predict stable rotating wave solutions in the Gray-Scott model. Specifically, we use local analysis in this section to predict stable rotating wave solutions corresponding to modes m = 1 and m = 2. There are essentially two ways that we can do this. In the case where a symmetry-breaking Hopf point is a primary bifurcation point, a center manifold/normal form reduction to (3.3) truncated at some order, coupled with the results in [Golubitsky and Roberts, 1987] permits us, in principle, to predict local branching and stability, including the unfolding of degeneracies up to codimension two. In this paper we concentrate on the codimension- one degeneracies and present examples of each of the four types.

'Y4. bg 4

2 H

sets and corrspondng toopen egion 4g an 4 boe

primary.~~~~~~~~~~~~~ Reut in' thi secio on the;m =0,

,

mR8

"'

4g-8#"5 4h-88 '8'#

M= -21 2'''''88 8'#''"#< -

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~fineato cosdrbl exen th paaee spac region. of stbl rottin

waveR-'s:.':--''2

..:R''-

For this model. Aurves mhoreng complated analysis bifurcationvHolv tHemI 1, m =2 eight- dimensionalandm=0,m=2six-dimensional Hopf-Hopf interactionsforsetsandC.Bottom:linearstabilt sugeAdstaes that sethe mnd =2 corotatoning wavopesn begionstbenwn 4g and4hpo

righ ofbthe other mh0and, me sa 1n Hop pons4n that the meini pa1randte mpc =2erotatn wvet can be simultnes Hopf bifurcation Both ofmr tese est are scsixsten aithpt-h

exeimentioal resultsp ineatof s [Norszticis et andl., 1987].: Thnearesuablits for ivrath mtea2yrotatingfo waets Bare weakr,sodn as suggestedoby the wordin of thapeiossetnc,thnths

fOrn=1 they dote hnot proe stability?in this stuatreion but pareamerely cosstaen wiher it Thmetybekn discussrcaion intiseiois orgaizedasy feomsvrsallos Fist we decrberesult the physically realistic case of nearly equal diffusion coefficients. Thigsmrin1 analysis of the Hopf-Hopf mode interactions to the forefront because it allows us to predict stable symmetry-breaking periodic solutions originating from Hopf points that are not primary. Results in this section on the ma=i0, mt=n1 six-dimensional Hopf-Hopf interaction considerably extend the parameter space region of stable rotating waves for this model. A more complicated analysis involving the m = 1, m = 2 eight- dimensional and m = 0, m = 2 six-dimensional Hopf-Hopf interactions suggests that the m = 2 rotating waves can be stable evheeraen the m = 2 Hopf point occurs to the right of both the m = 0 and m = 1 Hopf points and that the m = 1 and m = 2 rotating wave can be simultaneously stable. Both of these results are consistent with the experimental results of [Noszticzius et al., 1987]. The results for the m = 2 rotating waves are weaker, as suggested by the wording of the previous sentence, than those for m = 1; they do not prove stability in this situation but are merely consistent with it. The discussion in this section is organized as follows. First, we describe results for the three parameter sets A- C on stable rotating waves for modes m = 1 and m = 2 produced via primary bifurcations. In doing so, we encounter examples of each of the four codimension -one degenerate 0(2) Hopf bifurcations of [Golubitsky and Roberts, 1987]. Tracking these degeneracies produces additional curves, which further subdivide the parameter space regions in Figs. 8 and 10. Computation of higher-order normal form coefficients on these curves of degenerate 0(2) Hopf bifurcation permits us to use the results of [Golubitsky and Roberts, 1987] to